We study the small speed of light expansion of general relativity, utilizing the modern perspective on non-Lorentzian geometry. This is an expansion around the ultra-local Carroll limit, in which light cones close up. To this end, we first rewrite the Einstein-Hilbert action in pre-ultra-local variables, which is closely related to the 3+1 decomposition of general relativity. At leading order in the expansion, these pre-ultra-local variables yield Carroll geometry and the resulting action describes the electric Carroll limit of general relativity. We also obtain the next-to-leading order action in terms of Carroll geometry and next-to-leading order geometric fields. The leading order theory yields constraint and evolution equations, and we can solve the evolution analytically. We furthermore construct a Carroll version of Bowen-York initial data, which has associated conserved boundary linear and angular momentum charges. The notion of mass is not present at leading order and only enters at next-to-leading order. This is illustrated by considering a particular truncation of the next-to-leading order action, corresponding to the magnetic Carroll limit, where we find a solution that describes the Carroll limit of a Schwarzschild black hole. Finally, we comment on how a cosmological constant can be incorporated in our analysis.
Cited by 3
Ekiz et al., Non-relativistic and ultra-relativistic scaling limits of multimetric gravity
J. High Energ. Phys. 2022, 151 (2022) [Crossref]
Figueroa-O’Farrill et al., The gauging procedure and carrollian gravity
J. High Energ. Phys. 2022, 243 (2022) [Crossref]
Concha et al., Non-relativistic and ultra-relativistic expansions of three-dimensional spin-3 gravity theories
J. High Energ. Phys. 2022, 155 (2022) [Crossref]
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- 1 Eidgenössische Technische Hochschule Zürich / Swiss Federal Institute of Technology in Zurich (ETH) [ETH Zurich]
- 2 Niels Bohr Institute [NBI]
- 3 Nordisk Institut for Teoretisk Fysik / Nordic Institute for Theoretical Physics [NORDITA]
- 4 Princeton University